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Assignment: Add and Subtract Rational Expressions
Choose any four (4) of the following five problems to solve. Show all work leading to your answer.
6 6
represents the
x 3x
total time it takes Randy to drive to work in a day, where x represents the average speed
of the car in miles per hour, during the first part of the drive.
6 6
a. Use addition to simplify the expression
. Show all the steps necessary to
x 3x
write the expression with a common denominator and to simplify the expression.
1. Randy drives his car to work each day. Suppose the expression
(6/x) + (6/3x)
6/x * 3/3 = 18/3x
b. If+the
speed
of the car was 24 mph during the first part of the drive, find the total
18/3x
6/3x
= 24/3x
amount of time it took Randy to drive to work that day.
24/3x ; substituting and then simplifying:
( 24 / 3*24 ) = 1/3
It took Randy 1/3 of an hour, or 20 minutes to drive to work that day.
325
325
represents the difference between the times,
x
x 15
measured in hours, it took two moving trucks to drive 325 miles from one town to another,
where x represents the average speed of the slower truck during the drive.
325
325
c. Use subtraction to simplify the expression
. Show all the steps
x
x 15
necessary to write the expression with a common denominator and to simplify
the expression.
2. Suppose the expression
325/x - 325/(x + 15)
finding common denominators: { (325/x) * [ (x + 15) / (x + 15) ] } = (325x + 4875) / (x^2 + 15x)
and { [ 325/(x + 15) ] * (x/x) } = [ 325x / (x^2 + 15x) ] ; then substituting and simplifying:
[ (325x + 4875) / (x^2 + 15x) ] - [ 325x / (x^2 + 15x) ] =
d.(325x
If the
slower
truck
drove
at an=average speed of 50 mph, find the difference
+ 4875
- 325x)
/ (x^2
+ 15x)
between the
driving
times.
4875
/ (x^2
+ 15x)
4875 / (x^2 + 15x)
then substituting and simplifying: { 4875 / [ 50^2 + 15(50) ] } = { [ 4875 / (2500+750) ] }
= 4875/3250 = 3/2
The difference between the driving times of the faster and slower trucks is 3/2 of an hour, or
1 hour and 30 minutes.
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3. Two cement mixing trucks are scheduled to pour cement for the foundation of a new
apartment building. Suppose the expression
100
x
100
represents the total time
x 1
expected for both trucks to complete the job, where x represents the rate in gallons per
minute, at which the slower truck can pour concrete.
a. Use addition to simplify the expression
100
x
100
. Show all the steps
x 1
necessary to write the expression with a common denominator and to simplify
the expression.
(100/x) + [ 100/(x + 1) ]
finding common denominators: ( 100/x ) * [ (x + 1) / (x + 1) ] = (100x + 100)/(x^2 + x)
and [ 100/(x + 1) ] * ( x/x ) = 100x/(x^2 + x) ; then substituting and simplifying:
(100x + 100 + 100x ) / (x^2 + x) = (200x + 100)/(x^2 + x)
b. If the slower truck can pour concrete at a rate of 24 gallons per minute, find the
total amount of time it would take both cement mixing trucks to complete the job.
Using (200x + 100)/(x^2 + x); then substituting and simplifying;
we get: [ (200*24) + 100 ] / (24^2 + 24) = 4900 / 600 = 49/6 .
Total time it would take for both cement mixing trucks to complete the job is 49/6 minutes,
or 8 minutes and 10 seconds.
4. A cyclist had the wind at his back for part of his 25-mile ride, and then the wind died down
15
10
for the remainder of the ride. Suppose the expression
represents the total
x 8 x
time it took the cyclist to complete his ride, where x represents the cyclist’s average riding
speed.
15
10
. Show all the steps necessary
x 8 x
to write the expression with a common denominator and to simplify the
expression.
a. Use addition to simplify the expression
15/(x + 8) + 10/x
15/(x + 8) * x/x = 15x /(x^2 + 8x)
10/x * (x + 8)/(x + 8) = (10x + 80)/(x^2 + 8x)
b. If the cyclist’s rides at an average speed of 10 miles per hour, find the total time it
15x/(x^2
+ (10x +the
80)/(x^2
took +to8x)
complete
ride. + 8x) = (25x + 80)/(x^2 + 8x)
Using ((25x + 80)/(x^2 + 8x); then substituting and simplifying;
we get: 250 + 80 / 100 + 80 = 330/180 = 11/6 .
It took 11/6 of an hour, or 1 hour and 50 minutes to complete the ride.
5. A school of salmon swims 100 feet against the current in a river and then swims the
100 100
same distance through a stretch of still water. Suppose the expression
x 5
x
represents the difference between the times it takes the salmon to swim against the
current versus in the still water, where x represents the average speed of the salmon in
still water.
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100 100
. Show all the steps
x 5
x
necessary to write the expression with a common denominator and to simplify
the expression.
a. Use subtraction to simplify the expression
{[100/(x-5)]*[x/x]} - {[100/x]}*[(x-5)/(x-5)]} = [100x-100x+500]/[x(x-5)] = [500/(x^2-5x)]
b. Find the difference between the times if the average speed of the salmon is 20
feet per minute.
if x=20, then {500/[20^2-5(20)]} = 500/[400-100] = 500/300 = 5/3 = 1min 40sec
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